openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /ComplexShellEnergyEq.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Fourier.LowFrequencyKernel | |
| import OAI.MeasureTheory.Falconer.Fourier.MeasureKernelConvolution | |
| import OAI.MeasureTheory.Falconer.Graphs.SourceSpatialEnergy | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Shells.ShellKernels | |
| import OAI.MeasureTheory.Falconer.Shells.Shells | |
| import OAI.MeasureTheory.Falconer.Shells.SmoothShellPartition | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.InverseAdd | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter PlanarFalconer | |
| open scoped Topology ENNReal | |
| lemma complexShellEnergy_eq (ν : ComplexMeasure ℝ) (N : ℕ) : complexShellEnergy ν N = | |
| ∫ r in dyadicShell N, ‖complexChar ν r‖^2 := by | |
| have hs := (dyadicShell_compact N).measurableSet | |
| have he : (dyadicShell N).indicator (fun r => ‖complexChar ν r‖^2) = | |
| fun r => (dyadicShell N).indicator (fun r => ‖complexFourier ν r‖^2) (-r) := by | |
| funext r | |
| have hr : -r ∈ dyadicShell N ↔ r ∈ dyadicShell N := by simp [dyadicShell] | |
| by_cases hm : r ∈ dyadicShell N | |
| · simp [hm,hr.mpr hm,complexChar_eq_fourier] | |
| · simp [hm,show -r ∉ dyadicShell N from fun h => hm (hr.mp h)] | |
| rw [complexShellEnergy,←integral_indicator hs,←integral_indicator hs,he,integral_neg_eq_self] | |
| lemma complexShellEnergy_nonneg (ν : ComplexMeasure ℝ) (N : ℕ) : 0 ≤ complexShellEnergy ν N := | |
| integral_nonneg (fun _ => sq_nonneg _) | |
| lemma complexMultiplier_energy_le (ν : ComplexMeasure ℝ) (n : ℕ) : | |
| (∫ r, ‖complexMultiplier ν n r‖^2) ≤ complexShellEnergy ν (n+1) := by | |
| have he : (∫ r in dyadicShell (n+1), ‖complexMultiplier ν n r‖^2) = | |
| ∫ r, ‖complexMultiplier ν n r‖^2 := | |
| setIntegral_eq_integral_of_forall_compl_eq_zero (fun r hr => by | |
| rw [complexMultiplier_zero ν n r hr,norm_zero,zero_pow (by norm_num)]) | |
| rw [←he,complexShellEnergy_eq] | |
| apply setIntegral_mono_on | |
| (((complexMultiplier_continuous ν n).norm.pow 2).continuousOn.integrableOn_compact | |
| (dyadicShell_compact (n+1))) | |
| (((complexChar_continuous ν).norm.pow 2).continuousOn.integrableOn_compact | |
| (dyadicShell_compact (n+1))) (dyadicShell_compact (n+1)).measurableSet | |
| intro r hr | |
| apply pow_le_pow_left₀ (norm_nonneg _) _ 2 | |
| rw [complexMultiplier,norm_mul,Complex.norm_real,Real.norm_eq_abs] | |
| exact (mul_le_mul_of_nonneg_right (sourceShellBand_abs_le n r) (norm_nonneg _)).trans_eq (one_mul _) | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter PlanarFalconer | |
| open scoped Topology ENNReal FourierTransform SchwartzMap | |
| noncomputable def lowKernel (n : ℕ) (x : ℝ) : ℂ := | |
| (2:ℝ)^n • sourceLowKernel ((2:ℝ)^n*x) | |
| lemma lowKernel_continuous (n : ℕ) : Continuous (lowKernel n) := by | |
| unfold lowKernel sourceLowKernel | |
| exact (continuous_const (y := (2:ℝ)^n)).smul ((sourceInverseKernel_continuous _ sourceLowpassSchwartz.integrable).comp | |
| ((continuous_const (y := (2:ℝ)^n)).mul continuous_id)) | |
| lemma lowKernel_integrable (n : ℕ) : Integrable (lowKernel n) := by | |
| have hi := sourceLowKernel_integrable.comp_div (by positivity : ((2:ℝ)^n)⁻¹ ≠ 0) | |
| have he : (fun x => sourceLowKernel (x/((2:ℝ)^n)⁻¹)) = | |
| fun x => sourceLowKernel ((2:ℝ)^n*x) := by funext x;simp [div_eq_mul_inv,mul_comm] | |
| rw [he] at hi | |
| exact hi.smul ((2:ℝ)^n) | |
| lemma lowKernel_uniform (n : ℕ) : | |
| (∫ x, ‖lowKernel n x‖) = ∫ x, ‖sourceLowKernel x‖ := by | |
| simp_rw [lowKernel, norm_smul, Real.norm_eq_abs, abs_of_pos (by positivity : (0:ℝ)<2^n)] | |
| rw [integral_const_mul, Measure.integral_comp_mul_left (fun x => ‖sourceLowKernel x‖) ((2:ℝ)^n), | |
| smul_eq_mul,abs_of_pos (by positivity : (0:ℝ)<((2:ℝ)^n)⁻¹)] | |
| rw [←mul_assoc,mul_inv_cancel₀ (by positivity),one_mul] | |
| lemma lowKernel_inverse (n : ℕ) : lowKernel n = | |
| sourceInverseKernel (fun r => (sourceShellLowpass n r : ℂ)) := by | |
| funext x | |
| have h : (fun r => (sourceShellLowpass n r : ℂ)) = | |
| fun r => sourceLowpassSchwartz (r/(2:ℝ)^n) := by | |
| funext r | |
| simp [sourceShellLowpass] | |
| rw [h,sourceInverseKernel_dilate _ (by positivity),abs_of_pos (by positivity : (0:ℝ)<2^n)] | |
| rfl | |
| lemma shellKernel_eq_sub_low (n : ℕ) : sourceShellKernel n = lowKernel (n+1)-lowKernel n := by | |
| rw [lowKernel_inverse,lowKernel_inverse] | |
| have hi (m : ℕ) : Integrable (fun r => (sourceShellLowpass m r : ℂ)) := by | |
| convert sourceLowpassSchwartz.integrable.comp_div (by positivity : (2:ℝ)^m ≠ 0) using 1 | |
| funext r | |
| simp [sourceShellLowpass] | |
| have he : (sourceShellSchwartz n : ℝ → ℂ) = | |
| (fun r => (sourceShellLowpass (n+1) r : ℂ)) - (fun r => (sourceShellLowpass n r : ℂ)) := by | |
| funext r | |
| simp [sourceShellBand] | |
| funext x | |
| rw [sourceShellKernel,he,sourceInverseKernel] | |
| change (2*Real.pi)⁻¹ • (∫ r, Real.fourierChar (-inner ℝ r (x/(2*Real.pi))) • | |
| ((sourceShellLowpass (n+1) r : ℂ)-(sourceShellLowpass n r : ℂ))) = _ | |
| simp only [smul_sub] | |
| have hj (m : ℕ) : Integrable (fun r => Real.fourierChar (-inner ℝ r (x/(2*Real.pi))) • | |
| (sourceShellLowpass m r : ℂ)) := by | |
| exact (VectorFourier.fourierIntegral_convergent_iff Real.continuous_fourierChar | |
| (innerSL ℝ).continuous₂ (x/(2*Real.pi))).2 (hi m) | |
| rw [integral_sub (hj (n+1)) (hj n),smul_sub] | |
| rfl | |
| noncomputable def lowTerm (n : ℕ) (μ : Measure ℝ) [IsFiniteMeasure μ] : Lp (α:=ℝ) ℂ 1 volume := | |
| (memLp_one_iff_integrable.mpr (kernelMeasureConvolution_L1 (lowKernel n) | |
| (lowKernel_continuous n) (lowKernel_integrable n) μ).1).toLp | |
| (kernelMeasureConvolution (lowKernel n) μ) | |
| lemma lowTerm_norm_le (n : ℕ) (μ : Measure ℝ) [IsFiniteMeasure μ] : | |
| ‖lowTerm n μ‖ ≤ μ.real univ * (∫ x, ‖sourceLowKernel x‖) := by | |
| have hconvolution := kernelMeasureConvolution_L1 (lowKernel n) | |
| (lowKernel_continuous n) (lowKernel_integrable n) μ | |
| rw [lowTerm,Lp.norm_toLp,eLpNorm_one_eq_lintegral_enorm hconvolution.1.aestronglyMeasurable, | |
| ←integral_norm_eq_lintegral_enorm hconvolution.1.aestronglyMeasurable] | |
| simpa only [lowKernel_uniform] using hconvolution.2 | |
| lemma lowTerm_add (n : ℕ) (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν] : | |
| lowTerm n (μ+ν) = lowTerm n μ + lowTerm n ν := by | |
| apply Lp.ext | |
| have hμ := (kernelMeasureConvolution_prod_integrable (lowKernel n) | |
| (lowKernel_continuous n) (lowKernel_integrable n) μ).prod_right_ae | |
| have hν := (kernelMeasureConvolution_prod_integrable (lowKernel n) | |
| (lowKernel_continuous n) (lowKernel_integrable n) ν).prod_right_ae | |
| filter_upwards [hμ,hν,(by exact MemLp.coeFn_toLp _ : | |
| (lowTerm n (μ+ν) : ℝ → ℂ) =ᵐ[volume] kernelMeasureConvolution (lowKernel n) (μ+ν)), | |
| (by exact MemLp.coeFn_toLp _ : (lowTerm n μ : ℝ → ℂ) =ᵐ[volume] kernelMeasureConvolution (lowKernel n) μ), | |
| (by exact MemLp.coeFn_toLp _ : (lowTerm n ν : ℝ → ℂ) =ᵐ[volume] kernelMeasureConvolution (lowKernel n) ν), | |
| Lp.coeFn_add (lowTerm n μ) (lowTerm n ν)] with x hxμ hxν hxsum hxμv hxνv hxadd | |
| rw [hxadd,hxsum,Pi.add_apply,hxμv,hxνv] | |
| exact integral_add_measure hxμ hxν | |
| lemma lowTerm_sub_of_le (n : ℕ) (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (h : ν ≤ μ) : lowTerm n (μ-ν) = lowTerm n μ-lowTerm n ν := by | |
| have he := lowTerm_add n (μ-ν) ν | |
| simp only [Measure.sub_add_cancel_of_le h] at he | |
| exact eq_sub_iff_add_eq.mpr he.symm | |
| lemma lowTerm_zero (μ : Measure ℝ) [IsFiniteMeasure μ] : lowTerm 0 μ = sourceLowTerm ⟨μ,inferInstance⟩ := by | |
| have hk : lowKernel 0 = sourceLowKernel := by funext x;simp [lowKernel] | |
| unfold lowTerm sourceLowTerm | |
| congr 1 | |
| rw [hk] | |
| rfl | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| end OAI | |